Can neurons in MLP and filters in CNN be compared?
tl;dr The equivalent to a neuron in a Fully-Connected (FC) layer is the kernel (or filter) of a Convolution layer
Differences
The neurons of these two types of layers have two key differences. These are that the convolution layers implement:

Sparse connectivity, i.e. each neuron is connected only to an area of the input, not the whole.
Weight sharing, i.e. similar connections end up having the same weights. This is usually visualized as the same filter traversing the image.

Besides these two key differences, there are some other technical details, e.g. how the biases are implemented. Other than that they perform the same operation.
What causes some confusion is that the input of a CNN is usually 2 or 3-dimensional, while a FC is usually 1-dimensional. These aren't mandatory however. To better help visualize the differences between the two I made a couple of figures illustrating the differences between a conv-layer and a FC one, both in 1D.
Sparse connectivity
On the left are two FC neural networks. On the right, are layers with sparse connections.

Weight sharing
On the left is a sparsely connected network. The colors represent the different values of the weights. On the right is the same network with weight-sharing. Note that similar weights (i.e. arrows with the same direction in each layer) have the same value.


To answer your other questions:

Are filters not the same in the way that they convert an &quot;image&quot; to a new &quot;image&quot; based on the weights that are in that filter? And that the next layers use these new &quot;images&quot;?

Yes, if the input of a convolution layer is an image, so will the output. The next layer will also operate on an image.
However, I'd like to note that not all convolution layers accept images as their inputs. There are 1D and 3D convolutional layers as well.